Evaluate the integral $\int \sin ^{6} \frac{x}{2} \cos \frac{x}{2} d x$

Step 1 ：Let \(u = \sin(\frac{x}{2})\). Then, \(du = \frac{1}{2}\cos(\frac{x}{2})dx\). So, \(2du = \cos(\frac{x}{2})dx\).

Step 2 ：Substitute \(u\) and \(du\) into the integral. We get \(\int u^6 * 2du = 2\int u^6 du\).

Step 3 ：Now, we can easily integrate this to get \(\frac{2}{7}u^7 + C\).

Step 4 ：Substitute \(u\) back in to get the final answer: \(\boxed{\frac{2}{7}\sin^7(\frac{x}{2}) + C}\).